First, the fact that angles DXE and XYF are congruent means that they are identical angles. However, that alone is not enough to prove that the two lines are parallel, since line DX does not intersect both lines AE and FY. The first congruence relation just barely gives us what we need. Informal proof follows:
1. From the definition of a line, line AE has angle AXE of angle 180 degrees.
2. Since angles AXD and BXE are congruent, they form identical angles.
3. Since the angles AXD and DXE added together must form 180 degrees (from #1), then angle AXB must equal DXE (from #2).
From the definition of parallel lines, lines AE and FY must be parallel, since a line intersecting both of them results in identical angles.